We will calculate the required probabilities and values using the properties of the normal distribution and the cumulative Z-Score Table.

1. Probability of IQ greater than 119.5

To calculate this:

• Mean ($\mu$) = 100
• Standard Deviation ($\sigma$) = 15
• Value of interest ($X$) = 119.5

The Z-score is calculated as: $Z = \frac{X - \mu}{\sigma}$ $Z = \frac{119.5 - 100}{15} = \frac{19.5}{15} \approx 1.30$

Using a Z-Score Table:

• The cumulative probability for $Z = 1.30$ is approximately $0.9032$.
• The probability for $P(Z > 1.30)$ is: $P(Z > 1.30) = 1 - 0.9032 = 0.0968$

Convert to percent: $P(IQ > 119.5) = 9.68\%$

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2. Probability of IQ less than 95.5

Here:

• $X = 95.5$

Calculate the Z-score: $Z = \frac{X - \mu}{\sigma} = \frac{95.5 - 100}{15} = \frac{-4.5}{15} \approx -0.30$

Using a Z-Score Table:

• The cumulative probability for $Z = -0.30$ is approximately $0.3821$.

Convert to percent: $P(IQ < 95.5) = 38.21\%$

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3. Number of people with IQ less than 140.5 in a sample of 800

Here:

• $X = 140.5$

Calculate the Z-score: $Z = \frac{X - \mu}{\sigma} = \frac{140.5 - 100}{15} = \frac{40.5}{15} \approx 2.70$

Using a Z-Score Table:

• The cumulative probability for $Z = 2.70$ is approximately $0.9965$.

The number of people in the sample: $\text{Number} = 0.9965 \times 800 \approx 797$

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4. Number of people with IQ greater than 100 in a sample of 800

Here:

• The mean is $100$, so $Z = 0$.
• The cumulative probability for $Z = 0$ is $0.5000$.

The probability for $P(Z > 0)$ is: $P(Z > 0) = 1 - 0.5000 = 0.5000$

The number of people in the sample: $\text{Number} = 0.5000 \times 800 = 400$

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Final Answers

1. $P(IQ > 119.5) = 9.68\%$
2. $P(IQ < 95.5) = 38.21\%$
3. Number of people with IQ less than 140.5: $797$
4. Number of people with IQ greater than 100: $400$

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Do you want detailed explanations or need help with similar calculations?

Here are 5 related questions to explore:

1. What is the probability of having an IQ between 90 and 110?
2. How would the probabilities change if the standard deviation were 10 instead of 15?
3. What is the Z-score for an IQ of 130, and how do you interpret it?
4. In a sample of 1,000 people, how many would you expect to have an IQ between 85 and 115?
5. What percentile rank corresponds to an IQ of 105?

Tip: Always double-check the Z-Score Table to ensure accurate cumulative probabilities.